Answer :
- Divide the first term of the dividend by the first term of the divisor to find the first term of the quotient: $3x^4 / 3x = x^3$.
- Multiply the divisor by the first term of the quotient and subtract the result from the corresponding terms of the dividend.
- Repeat the process until all terms of the dividend have been used.
- The quotient is the sum of the terms found in each step: $\boxed{x^3 + 7x^2 - 1}$.
### Explanation
1. Understanding the Problem
We are asked to find the quotient when $3x^4 + 23x^3 + 14x^2 - 3x - 2$ is divided by $3x + 2$ using the box method. The box method is a visual representation of polynomial long division. We need to fill in the missing values in the box to find the quotient.
2. First Term of the Quotient
First, we set up the box method with the divisor $(3x + 2)$ on the side and the dividend inside the box. We start by dividing the first term of the dividend, $3x^4$, by the first term of the divisor, $3x$. This gives us $x^3$. We place $x^3$ above the first column. Then, we multiply the divisor $(3x + 2)$ by $x^3$ to get $3x^4 + 2x^3$, which we place in the first column of the box.
3. Finding the Remainder
Next, we subtract $3x^4 + 2x^3$ from $3x^4 + 23x^3$ to find the remaining term, which is $21x^3$. We bring down the next term from the dividend, $14x^2$, to get $21x^3 + 14x^2$.
4. Second Term of the Quotient
Now, we divide $21x^3$ by $3x$ to get $7x^2$. We place $7x^2$ above the second column. Then, we multiply the divisor $(3x + 2)$ by $7x^2$ to get $21x^3 + 14x^2$, which we place in the second column of the box.
5. Continuing the Process
We subtract $21x^3 + 14x^2$ from $21x^3 + 14x^2$ to get $0$. We bring down the next term from the dividend, $-3x$, to get $-3x$.
6. Third Term of the Quotient
We divide $-3x$ by $3x$ to get $-1$. We place $-1$ above the third column. Then, we multiply the divisor $(3x + 2)$ by $-1$ to get $-3x - 2$, which we place in the third column of the box.
7. Final Result
Finally, we subtract $-3x - 2$ from $-3x - 2$ to get $0$. The quotient is the sum of the terms above the box: $x^3 + 7x^2 - 1$.
8. Conclusion
Therefore, the quotient when $3x^4 + 23x^3 + 14x^2 - 3x - 2$ is divided by $3x + 2$ is $x^3 + 7x^2 - 1$.
### Examples
Polynomial division is used in various engineering fields, such as control systems, to analyze the stability and response of systems. For example, when designing a feedback control system, engineers use polynomial division to simplify transfer functions and determine the system's behavior. Understanding polynomial division helps engineers optimize system performance and ensure stability.
- Multiply the divisor by the first term of the quotient and subtract the result from the corresponding terms of the dividend.
- Repeat the process until all terms of the dividend have been used.
- The quotient is the sum of the terms found in each step: $\boxed{x^3 + 7x^2 - 1}$.
### Explanation
1. Understanding the Problem
We are asked to find the quotient when $3x^4 + 23x^3 + 14x^2 - 3x - 2$ is divided by $3x + 2$ using the box method. The box method is a visual representation of polynomial long division. We need to fill in the missing values in the box to find the quotient.
2. First Term of the Quotient
First, we set up the box method with the divisor $(3x + 2)$ on the side and the dividend inside the box. We start by dividing the first term of the dividend, $3x^4$, by the first term of the divisor, $3x$. This gives us $x^3$. We place $x^3$ above the first column. Then, we multiply the divisor $(3x + 2)$ by $x^3$ to get $3x^4 + 2x^3$, which we place in the first column of the box.
3. Finding the Remainder
Next, we subtract $3x^4 + 2x^3$ from $3x^4 + 23x^3$ to find the remaining term, which is $21x^3$. We bring down the next term from the dividend, $14x^2$, to get $21x^3 + 14x^2$.
4. Second Term of the Quotient
Now, we divide $21x^3$ by $3x$ to get $7x^2$. We place $7x^2$ above the second column. Then, we multiply the divisor $(3x + 2)$ by $7x^2$ to get $21x^3 + 14x^2$, which we place in the second column of the box.
5. Continuing the Process
We subtract $21x^3 + 14x^2$ from $21x^3 + 14x^2$ to get $0$. We bring down the next term from the dividend, $-3x$, to get $-3x$.
6. Third Term of the Quotient
We divide $-3x$ by $3x$ to get $-1$. We place $-1$ above the third column. Then, we multiply the divisor $(3x + 2)$ by $-1$ to get $-3x - 2$, which we place in the third column of the box.
7. Final Result
Finally, we subtract $-3x - 2$ from $-3x - 2$ to get $0$. The quotient is the sum of the terms above the box: $x^3 + 7x^2 - 1$.
8. Conclusion
Therefore, the quotient when $3x^4 + 23x^3 + 14x^2 - 3x - 2$ is divided by $3x + 2$ is $x^3 + 7x^2 - 1$.
### Examples
Polynomial division is used in various engineering fields, such as control systems, to analyze the stability and response of systems. For example, when designing a feedback control system, engineers use polynomial division to simplify transfer functions and determine the system's behavior. Understanding polynomial division helps engineers optimize system performance and ensure stability.
